1. S. Kurtek 1, E. Klassen 2, Z. Ding 3, A. Srivastava 1 1 Florida State University Department of Statistics 2 Florida State University Department of Mathematics 3 Vanderbilt University Institute of Imaging Science A NOVEL RIEMANNIAN FRAMEWORK FOR SHAPE ANALYSIS OF 3D OBJECTS *This research was supported in part by grants from AFOSR, ONR and NSF.

2. PROBLEM INTRODUCTION Main Goal: To compare the shapes of these surfaces using a metric that is invariant to scale, translation, rotation and re-parameterization. f1f1 f2f2 d(f 1,f 2 ) = ? Consider these two parameterized surfaces:

3. MOTIVATIONMOTIVATION 1.Medical Image Analysis 2.Bioinformatics 3.Facial Recognition 4.Geology 5.Image Matching

4. These do not analyze shapes of parameterized surfaces directly, which is our goal. CURRENT METHODS 1.Deformable Templates - Davatzikos et al. 1996; Joshi et al. 1997; Grenander and Miller 1998; Csernansky et al. 2002. 2.Level Set Methods - Malladi et al. 1996. 3.Landmarks, Active Shape Models - Kendall 1985; Cootes et al. 1995; Dryden and Mardia 1998. 4.Iterative Closest Point Algorithm - Besl and McKay 1992; Almhdie et al. 2007. 5.Medial Representation - Siddiqi and Pizer 1992; Bouix et al. 2001; Gorczowski et al. 2010.

5. PARAMETERIZED SURFACES MAIN ISSUE S denotes a 2D smooth and differentiable surface. Define a parameterization of surface S as. Let Г be the set of all diffeomorphisms of. The natural action of Г on is on the right by composition. In general, is not area preserving and therefore the isometry condition is not satisfied under the metric: Existing Solutions: 1.Restrict to area preserving re-parameterizations - Gu et al. 2004. 2.Fix the parameterization (SPHARM) of all surfaces - Brechbüler et al. 1995; Styner et al. 2006.

6. Definition: a.Given a differentiable surface f, define as the “area multiplication factor” of f at s: where {u s, v s } is an orthonormal basis of. b. Define a q-map, using by NEW REPRESENTATION OF SURFACES

7. SHAPE ANALYSIS OF SURFACES Achieving the desired invariances: Remove Directly: 1.Scale,. 2.Translation,. Remove Using Algebraic Operations: 1.Rotation, SO(3): Given, the action of the rotation group is defined as (O,q)=Oq. 2.Re-parameterization, Г: Given, the action of the re- parameterization group is defined as

8. Equivalence Class: Shape Space: Distance Between Surfaces: Distance Between Orbits: DISTANCE BETWEEN SURFACES Optimization Problem: 1.Rotation, SO(3): Procrustes analysis. 2.Re-Parameterization, Г: gradient descent.

9. 1.Define the energy as: where γ 0 is fixed and γ is a variable. 2.Define the mapping: 3.The Jacobian of φ(γ) is: where b is an orthonormal basis of and. 4.The directional derivative of E is: OPTIMIZATION PROBLEM OVER Г

10. Optimize over 60 elements in the group of symmetries of the dodecahedron. The largest finite subgroup of SO(3). Equivalent to placing the North Pole at 60 different positions. INITIALIZATION OF GRADIENT SEARCH f1f1 Energy Minimizerf2f2 Cost Function

11. BRAIN STRUCTURE SURFACES TWO LEFT PUTAMENS BRAIN STRUCTURE SURFACES TWO LEFT PUTAMENS f1f1 f2f2 O*(f 2 ◦ γ*) Energy at each iteration ||γ*(s)-s|| d([q 1 ],[q 2 ])= =0.0207

12. BRAIN STRUCTURE SURFACES LEFT PUTAMEN AND LEFT THALAMUS BRAIN STRUCTURE SURFACES LEFT PUTAMEN AND LEFT THALAMUS f1f1 f2f2 O*(f 2 ◦ γ*) Energy at each iteration d([q 1 ],[q 2 ])= =0.0790 ||γ*(s)-s||

13. T1 weighted brain magnetic resonance images of young adults of ages between 13 and 17 recruited from the Detroit Fetal Alcohol and Drug Exposure Cohort. Left and right brain structures (total of 11) for 34 subjects, 19 with ADHD and 15 healthy. Leave-one-out nearest neighbor classification scheme. ADHD STUDY SINGLE STRUCTURE CLASSIFICATION

14. ADHD STUDY MULTIPLE STRUCTURE CLASSIFICATION Combined weighted single structure distances to maximize the ADHD classification rate. Using our method, the combination of left putamen, left pallidus and right pallidus distances provided a 91.2% classification rate. Other methods: 1.Harmonic – 85.3% 2.ICP – 88.2% 3.SPHARM-PDM – 85.3%

15. So far we have presented the framework and results for shape analysis of closed surfaces only. The extension to quadrilateral (D=[0,1] 2 ) and hemispherical (D=unit disk) surfaces is straightforward. EXTENSION TO OTHER TYPES OF SURFACES

16. QUADRILATERAL SURFACES IMAGE MATCHING QUADRILATERAL SURFACES IMAGE MATCHING f1f1 f2f2 O*(f 2 ◦ γ*) γ* Energy at each iteration I1I1 I2I2 |I 1 -I 2 ||I 1 -O*(I 2 ◦ γ*)| d([q 1 ],[q 2 ])= =0.0567

17. QUADRILATERAL SURFACES IMAGE MATCHING QUADRILATERAL SURFACES IMAGE MATCHING f1f1 f2f2 O*(f 2 ◦ γ*) γ* Energy at each iteration I1I1 I2I2 |I 1 -I 2 ||I 1 -O*(I 2 ◦ γ*)| d([q 1 ],[q 2 ])= =0.0953

18. HEMISPHERICAL SURFACES CROPPED FACES HEMISPHERICAL SURFACES CROPPED FACES f1f1 f2f2 O*(f 2 ◦ γ*) Energy at each iteration d([q 1 ],[q 2 ])= =0.0288 ||γ*(s)-s||

19. OPTIMAL PATHS BETWEEN SURFACES Without Re-Parameterization With Re-Parameterization We computed certain optimal paths between two toy shapes with and without re-parameterization. The displayed paths are not geodesic with respect to our metric but under a metric described by Kilian et al. 2007. M. Kilian, N. Mitra, and H. Pottman. “Geometric Modeling in Shape Space.”, in ACM Transactions on Graphics, vol. 26, no. 3, 2007, 1-8.

20. CONCLUSION AND FUTURE WORK Shape analysis of 3D objects is very important in many scientific fields. We have proposed a novel approach for the analysis of 3D objects, which is invariant to rigid motion, scaling and most importantly re-parameterization. This results in a proper metric on the space of q-maps. In the future, we would like to be able to show geodesics between surfaces. We would also like to apply this methodology to more data sets.

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